Risk

Risk and uncertainty are words that relate to one another and have different meanings to different people. Even dictionary definitions are at odds with common usage and understanding. That being said, we will define uncertainty and risk as follows:

**Uncertainty** is a state of having limited knowledge where it
is impossible to exactly describe the existing state or future outcomes.
Information is not complete and expected
values cannot be determined. If we can derive probabilities for
each possible outcome, as will be the focus of this experiment, we can then
compute expected values. For any individual investment or portfolio of
investments, we can compute the expected values of the 'net' of gains and
losses-- which we will just call the expected value, as well as the expected
value of losses, which is defined as the **Risk**.

If we invest in a large number of opportunities with positive expected values over a long period of time, we can rely on the law of large numbers to hold such that the actual net gains will be very close to the expected values. In such cases, the risk, or expected value of the loss, will be very small. However, when we can't rely on the law of large numbers -- it is common practice to talk about the risk vs. the reward, meaning the expected value of the loss, vs. the expected value of the net gain minus loss.

Consider the case where you had two investment opportunities, each with positive expected values. If the opportunities were negatively correlated -- meaning that when the outcome for one is higher than its expected value, the outcome for the other tends to be lower than its expected value. Suppose that instead of investing a given amount in either of the investments, you consider diversifying by investing 1/2 the amount in each of the two negatively correlated investments, what is likely to happen? One of the investments, we don't know which, will have an outcome higher than its expected value and the other will have one lower than its expected value and thus the expected value of the combined investment will be less than that of the expected value of investing the entire amount in either of the investments. On the other hand, if one of the investments were to lose money, the other investment which is negatively correlated would tend to make money and so the expected value of the loss, or risk, would be lower.

Consider the Capital One LEAP options purchased by Jamie Mai and Charlie Ledley, as described in The Big Short. They had reasoned that Capital One stocks, then selling at about $30 per share, would either drop close to zero or increase to $60 or more, and had purchased options to buy Capital One shares at $40 per share, hoping to gain when the price of the stock rose. They could have lessened their risk of taking a loss (in case Capital One was found guilty and their shares dropped instead of going up) by purchasing options to sell Capital One shares near zero in addition to buying options to purchase shares at 40. If they had done this, they would have made only half as much. However, if Capital One had been found guilty and their price had dropped close to zero, they would still have made a sizeable profit.